The Postman's Hat is one of the masks (or should they say hat?) in The Legend of Zelda: Majora's Mask. You basically get it after helping out the Postman, and he leaves Termina on a happy note.
The only real power it has is that it gives Link to ability to look into maiboxes. One for a piece of heart, the other times it just spits out rupees. Who puts currency in a mail box?